| L(s) = 1 | − 3-s − 2·5-s + 9-s + 4·11-s + 2·15-s − 2·17-s − 4·19-s − 25-s − 27-s − 2·29-s − 4·33-s + 2·37-s − 6·41-s + 4·43-s − 2·45-s − 8·47-s + 2·51-s − 10·53-s − 8·55-s + 4·57-s − 4·59-s + 2·61-s + 4·67-s + 2·73-s + 75-s + 8·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s + 1.20·11-s + 0.516·15-s − 0.485·17-s − 0.917·19-s − 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.696·33-s + 0.328·37-s − 0.937·41-s + 0.609·43-s − 0.298·45-s − 1.16·47-s + 0.280·51-s − 1.37·53-s − 1.07·55-s + 0.529·57-s − 0.520·59-s + 0.256·61-s + 0.488·67-s + 0.234·73-s + 0.115·75-s + 0.900·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5103532124\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5103532124\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.91935324317456, −12.52530565662164, −12.09739496833942, −11.54215203382722, −11.33400838684112, −10.89727758857358, −10.35108725512108, −9.698166094506648, −9.370579698393474, −8.762938135291325, −8.268136813604129, −7.875665457306942, −7.193676647818878, −6.818720011295170, −6.293007014302012, −5.965533834368151, −5.184322869497133, −4.577112272674247, −4.284908358998643, −3.664693990813824, −3.301302421932513, −2.355866556975317, −1.735499310260932, −1.120630312779711, −0.2269549421159036,
0.2269549421159036, 1.120630312779711, 1.735499310260932, 2.355866556975317, 3.301302421932513, 3.664693990813824, 4.284908358998643, 4.577112272674247, 5.184322869497133, 5.965533834368151, 6.293007014302012, 6.818720011295170, 7.193676647818878, 7.875665457306942, 8.268136813604129, 8.762938135291325, 9.370579698393474, 9.698166094506648, 10.35108725512108, 10.89727758857358, 11.33400838684112, 11.54215203382722, 12.09739496833942, 12.52530565662164, 12.91935324317456
